The hypoelliptic Laplacian on the cotangent bundle

Bismut, Jean-Michel¹

¹ Département de Mathématique, Université Paris-Sud, Bâtiment 425, 91405 Orsay, France

Journal of the American Mathematical Society, Tome 18 (2005) no. 2, pp. 379-476

Résumé

In this paper, we construct a new version of Hodge theory, where the corresponding Laplacian acts on the total space of the cotangent bundle. This Laplacian is a hypoelliptic operator, which is in general non-self-adjoint. When properly interpreted, it provides an interpolation between classical Hodge theory and the generator of the geodesic flow. The construction is also done in families in the superconnection formalism of Quillen and extends earlier work by Lott and the author.

Détail

DOI : 10.1090/S0894-0347-05-00479-0

Bibliographie

[1] Atiyah, M. F. Circular symmetry and stationary-phase approximation Astérisque 1985 43 59

[2] Berline, Nicole, Getzler, Ezra, Vergne, Michã¨Le Heat kernels and Dirac operators 1992

[3] Berthomieu, Alain, Bismut, Jean-Michel Quillen metrics and higher analytic torsion forms J. Reine Angew. Math. 1994 85 184

[4] Bismut, Jean-Michel Mécanique aléatoire 1981

[5] Bismut, Jean-Michel Index theorem and equivariant cohomology on the loop space Comm. Math. Phys. 1985 213 237

[6] Bismut, Jean-Michel The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs Invent. Math. 1986 91 151

[7] Bismut, J.-M. Bott-Chern currents, excess normal bundles and the Chern character Geom. Funct. Anal. 1992 285 340

[8] Bismut, Jean-Michel On certain infinite-dimensional aspects of Arakelov intersection theory Comm. Math. Phys. 1992 217 248

[9] Bismut, Jean-Michel Equivariant immersions and Quillen metrics J. Differential Geom. 1995 53 157

[10] Bismut, Jean-Michel Holomorphic families of immersions and higher analytic torsion forms Astérisque 1997

[11] Bismut, Jean-Michel Holomorphic and de Rham torsion Compos. Math. 2004 1302 1356

[12] Bismut, Jean-Michel Une déformation de la théorie de Hodge sur le fibré cotangent C. R. Math. Acad. Sci. Paris 2004 471 476

[13] Bismut, Jean-Michel Le laplacien hypoelliptique sur le fibré cotangent C. R. Math. Acad. Sci. Paris 2004 555 559

[14] Bismut, Jean-Michel Une déformation en famille du complexe de de Rham-Hodge C. R. Math. Acad. Sci. Paris 2004 623 627

[15] Bismut, Jean-Michel, Freed, Daniel S. The analysis of elliptic families. I. Metrics and connections on determinant bundles Comm. Math. Phys. 1986 159 176

[16] Bismut, J.-M., Gillet, H., Soulã©, C. Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion Comm. Math. Phys. 1988 49 78

[17] Bismut, Jean-Michel, Goette, Sebastian Families torsion and Morse functions Astérisque 2001

[18] Bismut, Jean-Michel, Goette, Sebastian Equivariant de Rham torsions Ann. of Math. (2) 2004 53 216

[19] Bismut, Jean-Michel, Lebeau, Gilles Complex immersions and Quillen metrics Inst. Hautes Études Sci. Publ. Math. 1991

[20] Bismut, Jean-Michel, Lott, John Flat vector bundles, direct images and higher real analytic torsion J. Amer. Math. Soc. 1995 291 363

[21] Bismut, Jean-Michel, Zhang, Weiping An extension of a theorem by Cheeger and Müller Astérisque 1992 235

[22] Bismut, J.-M., Zhang, W. Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle Geom. Funct. Anal. 1994 136 212

[23] Bott, Raoul The stable homotopy of the classical groups Ann. of Math. (2) 1959 313 337

[24] Cheeger, Jeff Analytic torsion and the heat equation Ann. of Math. (2) 1979 259 322

[25] Fried, David The zeta functions of Ruelle and Selberg. I Ann. Sci. École Norm. Sup. (4) 1986 491 517

[26] Fried, David Torsion and closed geodesics on complex hyperbolic manifolds Invent. Math. 1988 31 51

[27] Helffer, B., Sjã¶Strand, J. Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten Comm. Partial Differential Equations 1985 245 340

[28] Hã¶Rmander, Lars Hypoelliptic second order differential equations Acta Math. 1967 147 171

[29] Knudsen, Finn Faye, Mumford, David The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div” Math. Scand. 1976 19 55

[30] Kolmogoroff, A. Zufällige Bewegungen (zur Theorie der Brownschen Bewegung) Ann. of Math. (2) 1934 116 117

[31] Mathai, Varghese, Quillen, Daniel Superconnections, Thom classes, and equivariant differential forms Topology 1986 85 110

[32] Moscovici, Henri, Stanton, Robert J. 𝑅-torsion and zeta functions for locally symmetric manifolds Invent. Math. 1991 185 216

[33] Cheeger, Jeff Analytic torsion and the heat equation Ann. of Math. (2) 1979 259 322

[34] Quillen, Daniel Superconnections and the Chern character Topology 1985 89 95

[35] Kvillen, D. Determinants of Cauchy-Riemann operators on Riemann surfaces Funktsional. Anal. i Prilozhen. 1985

[36] Ray, D. B., Singer, I. M. 𝑅-torsion and the Laplacian on Riemannian manifolds Advances in Math. 1971 145 210

[37] Witten, Edward Supersymmetry and Morse theory J. Differential Geometry 1982